Average Error: 16.3 → 8.1
Time: 24.9s
Precision: 64
\[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(y + x\right) - \frac{\left(z - t\right) \cdot y}{a - t} = -\infty:\\ \;\;\;\;\mathsf{fma}\left(y, \left(\frac{\frac{t}{\sqrt[3]{a - t}}}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} - \frac{z}{a - t}\right) + 0 \cdot \frac{z}{a - t}, y + x\right)\\ \mathbf{elif}\;\left(y + x\right) - \frac{\left(z - t\right) \cdot y}{a - t} \le -3.318045847118545880487936801930202223482 \cdot 10^{-266}:\\ \;\;\;\;\left(y + x\right) - \frac{\left(z - t\right) \cdot y}{a - t}\\ \mathbf{elif}\;\left(y + x\right) - \frac{\left(z - t\right) \cdot y}{a - t} \le 4.461734320096487882787345408186744745213 \cdot 10^{-218}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \left(\frac{1}{a - t} \cdot t - \frac{z}{a - t}\right) + 0 \cdot \frac{z}{a - t}, y + x\right)\\ \end{array}\]
\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}
↓
\begin{array}{l}
\mathbf{if}\;\left(y + x\right) - \frac{\left(z - t\right) \cdot y}{a - t} = -\infty:\\
\;\;\;\;\mathsf{fma}\left(y, \left(\frac{\frac{t}{\sqrt[3]{a - t}}}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} - \frac{z}{a - t}\right) + 0 \cdot \frac{z}{a - t}, y + x\right)\\

\mathbf{elif}\;\left(y + x\right) - \frac{\left(z - t\right) \cdot y}{a - t} \le -3.318045847118545880487936801930202223482 \cdot 10^{-266}:\\
\;\;\;\;\left(y + x\right) - \frac{\left(z - t\right) \cdot y}{a - t}\\

\mathbf{elif}\;\left(y + x\right) - \frac{\left(z - t\right) \cdot y}{a - t} \le 4.461734320096487882787345408186744745213 \cdot 10^{-218}:\\
\;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(y, \left(\frac{1}{a - t} \cdot t - \frac{z}{a - t}\right) + 0 \cdot \frac{z}{a - t}, y + x\right)\\

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r13154523 = x;
        double r13154524 = y;
        double r13154525 = r13154523 + r13154524;
        double r13154526 = z;
        double r13154527 = t;
        double r13154528 = r13154526 - r13154527;
        double r13154529 = r13154528 * r13154524;
        double r13154530 = a;
        double r13154531 = r13154530 - r13154527;
        double r13154532 = r13154529 / r13154531;
        double r13154533 = r13154525 - r13154532;
        return r13154533;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r13154534 = y;
        double r13154535 = x;
        double r13154536 = r13154534 + r13154535;
        double r13154537 = z;
        double r13154538 = t;
        double r13154539 = r13154537 - r13154538;
        double r13154540 = r13154539 * r13154534;
        double r13154541 = a;
        double r13154542 = r13154541 - r13154538;
        double r13154543 = r13154540 / r13154542;
        double r13154544 = r13154536 - r13154543;
        double r13154545 = -inf.0;
        bool r13154546 = r13154544 <= r13154545;
        double r13154547 = cbrt(r13154542);
        double r13154548 = r13154538 / r13154547;
        double r13154549 = r13154547 * r13154547;
        double r13154550 = r13154548 / r13154549;
        double r13154551 = r13154537 / r13154542;
        double r13154552 = r13154550 - r13154551;
        double r13154553 = 0.0;
        double r13154554 = r13154553 * r13154551;
        double r13154555 = r13154552 + r13154554;
        double r13154556 = fma(r13154534, r13154555, r13154536);
        double r13154557 = -3.318045847118546e-266;
        bool r13154558 = r13154544 <= r13154557;
        double r13154559 = 4.461734320096488e-218;
        bool r13154560 = r13154544 <= r13154559;
        double r13154561 = r13154537 / r13154538;
        double r13154562 = fma(r13154561, r13154534, r13154535);
        double r13154563 = 1.0;
        double r13154564 = r13154563 / r13154542;
        double r13154565 = r13154564 * r13154538;
        double r13154566 = r13154565 - r13154551;
        double r13154567 = r13154566 + r13154554;
        double r13154568 = fma(r13154534, r13154567, r13154536);
        double r13154569 = r13154560 ? r13154562 : r13154568;
        double r13154570 = r13154558 ? r13154544 : r13154569;
        double r13154571 = r13154546 ? r13154556 : r13154570;
        return r13154571;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original16.3
Target8.5
Herbie8.1
\[\begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt -1.366497088939072697550672266103566343531 \cdot 10^{-7}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \mathbf{elif}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt 1.475429344457723334351036314450840066235 \cdot 10^{-239}:\\ \;\;\;\;\frac{y \cdot \left(a - z\right) - x \cdot t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if (- (+ x y) (/ (* (- z t) y) (- a t))) < -inf.0

    1. Initial program 64.0

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\]

    2. Simplified28.6

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{t - z}{a - t}, y + x\right)}\]
    3. Using strategy rm

    4. Applied div-sub28.6

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{t}{a - t} - \frac{z}{a - t}}, y + x\right)\]
    5. Using strategy rm

    6. Applied add-cube-cbrt28.8

      \[\leadsto \mathsf{fma}\left(y, \frac{t}{a - t} - \color{blue}{\left(\sqrt[3]{\frac{z}{a - t}} \cdot \sqrt[3]{\frac{z}{a - t}}\right) \cdot \sqrt[3]{\frac{z}{a - t}}}, y + x\right)\]

    7. Applied add-cube-cbrt28.7

      \[\leadsto \mathsf{fma}\left(y, \frac{t}{\color{blue}{\left(\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}\right) \cdot \sqrt[3]{a - t}}} - \left(\sqrt[3]{\frac{z}{a - t}} \cdot \sqrt[3]{\frac{z}{a - t}}\right) \cdot \sqrt[3]{\frac{z}{a - t}}, y + x\right)\]

    8. Applied *-un-lft-identity28.7

      \[\leadsto \mathsf{fma}\left(y, \frac{\color{blue}{1 \cdot t}}{\left(\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}\right) \cdot \sqrt[3]{a - t}} - \left(\sqrt[3]{\frac{z}{a - t}} \cdot \sqrt[3]{\frac{z}{a - t}}\right) \cdot \sqrt[3]{\frac{z}{a - t}}, y + x\right)\]

    9. Applied times-frac28.8

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{1}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \frac{t}{\sqrt[3]{a - t}}} - \left(\sqrt[3]{\frac{z}{a - t}} \cdot \sqrt[3]{\frac{z}{a - t}}\right) \cdot \sqrt[3]{\frac{z}{a - t}}, y + x\right)\]

    10. Applied prod-diff28.8

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(\frac{1}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}, \frac{t}{\sqrt[3]{a - t}}, -\sqrt[3]{\frac{z}{a - t}} \cdot \left(\sqrt[3]{\frac{z}{a - t}} \cdot \sqrt[3]{\frac{z}{a - t}}\right)\right) + \mathsf{fma}\left(-\sqrt[3]{\frac{z}{a - t}}, \sqrt[3]{\frac{z}{a - t}} \cdot \sqrt[3]{\frac{z}{a - t}}, \sqrt[3]{\frac{z}{a - t}} \cdot \left(\sqrt[3]{\frac{z}{a - t}} \cdot \sqrt[3]{\frac{z}{a - t}}\right)\right)}, y + x\right)\]

    11. Simplified28.5

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\left(\frac{1 \cdot \frac{t}{\sqrt[3]{a - t}}}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} - \frac{z}{a - t}\right)} + \mathsf{fma}\left(-\sqrt[3]{\frac{z}{a - t}}, \sqrt[3]{\frac{z}{a - t}} \cdot \sqrt[3]{\frac{z}{a - t}}, \sqrt[3]{\frac{z}{a - t}} \cdot \left(\sqrt[3]{\frac{z}{a - t}} \cdot \sqrt[3]{\frac{z}{a - t}}\right)\right), y + x\right)\]

    12. Simplified28.5

      \[\leadsto \mathsf{fma}\left(y, \left(\frac{1 \cdot \frac{t}{\sqrt[3]{a - t}}}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} - \frac{z}{a - t}\right) + \color{blue}{\frac{z}{a - t} \cdot 0}, y + x\right)\]

    if -inf.0 < (- (+ x y) (/ (* (- z t) y) (- a t))) < -3.318045847118546e-266

    1. Initial program 1.4

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\]

    if -3.318045847118546e-266 < (- (+ x y) (/ (* (- z t) y) (- a t))) < 4.461734320096488e-218

    1. Initial program 54.9

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\]

    2. Simplified54.2

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{t - z}{a - t}, y + x\right)}\]

    3. Taylor expanded around inf 19.9

      \[\leadsto \color{blue}{\frac{z \cdot y}{t} + x}\]

    4. Simplified19.8

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}\]

    if 4.461734320096488e-218 < (- (+ x y) (/ (* (- z t) y) (- a t)))

    1. Initial program 11.9

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\]

    2. Simplified7.4

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{t - z}{a - t}, y + x\right)}\]
    3. Using strategy rm

    4. Applied div-sub7.4

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{t}{a - t} - \frac{z}{a - t}}, y + x\right)\]
    5. Using strategy rm

    6. Applied add-cube-cbrt7.6

      \[\leadsto \mathsf{fma}\left(y, \frac{t}{a - t} - \color{blue}{\left(\sqrt[3]{\frac{z}{a - t}} \cdot \sqrt[3]{\frac{z}{a - t}}\right) \cdot \sqrt[3]{\frac{z}{a - t}}}, y + x\right)\]

    7. Applied div-inv7.6

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{t \cdot \frac{1}{a - t}} - \left(\sqrt[3]{\frac{z}{a - t}} \cdot \sqrt[3]{\frac{z}{a - t}}\right) \cdot \sqrt[3]{\frac{z}{a - t}}, y + x\right)\]

    8. Applied prod-diff7.6

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(t, \frac{1}{a - t}, -\sqrt[3]{\frac{z}{a - t}} \cdot \left(\sqrt[3]{\frac{z}{a - t}} \cdot \sqrt[3]{\frac{z}{a - t}}\right)\right) + \mathsf{fma}\left(-\sqrt[3]{\frac{z}{a - t}}, \sqrt[3]{\frac{z}{a - t}} \cdot \sqrt[3]{\frac{z}{a - t}}, \sqrt[3]{\frac{z}{a - t}} \cdot \left(\sqrt[3]{\frac{z}{a - t}} \cdot \sqrt[3]{\frac{z}{a - t}}\right)\right)}, y + x\right)\]

    9. Simplified7.4

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\left(\frac{1}{a - t} \cdot t - \frac{z}{a - t}\right)} + \mathsf{fma}\left(-\sqrt[3]{\frac{z}{a - t}}, \sqrt[3]{\frac{z}{a - t}} \cdot \sqrt[3]{\frac{z}{a - t}}, \sqrt[3]{\frac{z}{a - t}} \cdot \left(\sqrt[3]{\frac{z}{a - t}} \cdot \sqrt[3]{\frac{z}{a - t}}\right)\right), y + x\right)\]

    10. Simplified7.4

      \[\leadsto \mathsf{fma}\left(y, \left(\frac{1}{a - t} \cdot t - \frac{z}{a - t}\right) + \color{blue}{\frac{z}{a - t} \cdot 0}, y + x\right)\]
  3. Recombined 4 regimes into one program.

  4. Final simplification8.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y + x\right) - \frac{\left(z - t\right) \cdot y}{a - t} = -\infty:\\ \;\;\;\;\mathsf{fma}\left(y, \left(\frac{\frac{t}{\sqrt[3]{a - t}}}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} - \frac{z}{a - t}\right) + 0 \cdot \frac{z}{a - t}, y + x\right)\\ \mathbf{elif}\;\left(y + x\right) - \frac{\left(z - t\right) \cdot y}{a - t} \le -3.318045847118545880487936801930202223482 \cdot 10^{-266}:\\ \;\;\;\;\left(y + x\right) - \frac{\left(z - t\right) \cdot y}{a - t}\\ \mathbf{elif}\;\left(y + x\right) - \frac{\left(z - t\right) \cdot y}{a - t} \le 4.461734320096487882787345408186744745213 \cdot 10^{-218}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \left(\frac{1}{a - t} \cdot t - \frac{z}{a - t}\right) + 0 \cdot \frac{z}{a - t}, y + x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019192 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTick from plot-0.2.3.4, B"

  :herbie-target
  (if (< (- (+ x y) (/ (* (- z t) y) (- a t))) -1.3664970889390727e-07) (- (+ y x) (* (* (- z t) (/ 1.0 (- a t))) y)) (if (< (- (+ x y) (/ (* (- z t) y) (- a t))) 1.4754293444577233e-239) (/ (- (* y (- a z)) (* x t)) (- a t)) (- (+ y x) (* (* (- z t) (/ 1.0 (- a t))) y))))

  (- (+ x y) (/ (* (- z t) y) (- a t))))